In this approach, one considers the trajectories in the space of unitaries such that the
interesting trajectory satisfies \(U(\tau =0)=\mathbb {1}\)  and \(U(\tau =1)=U\) . Further, one identifies the optimal path
or circuit, by minimising a cost functional defined for various possible paths.
These cost functionals satisfy certain physicality conditions, and then these can be shown to define
length functionals on a Riemannian manifold that arises in the space of unitary transformations,
for a specific choice of the cost functional [1]}, [2]}. The metric on the space of unitaries is written as
\(ds^{2} &=& 4\delta _{ab}\Big [\frac{1}{{\rm Tr} \big [J^{a}.J_{a}^{T}\big ]}\text{Tr}\Big (\partial _{\tau }{U(\tau )}\cdot U^{-1}(\tau )\cdot (J^{a})^{T}\Big )\Big ]\times \nonumber \\ &~&\Big [\frac{1}{{\rm Tr} \big [J^{b}.J_{b}^{T}\big ]}\text{Tr}\Big (\partial _{\tau }{U(\tau )}\cdot U^{-1}(\tau )\cdot (J^{b})^{T}\Big )\Big ]^{*}~,\) 
